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234                      ANALYTICAL MECHANICS
VQ < aooQ. Then the sliding at the point of contact is toward the left; therefore F is directed to the right and consequently positive. Thus
dv                                                       ,A A
n = M>                                         (4)
dco _     fig                                                          ,,,.
dt~~^                                      (5)
Integrating the last two equations we have
v = vQ + vgt*                                          (6)
 = wo-^.                                        (7)
These equations hold until sliding stops, after which the hoop rolls with constant angular and linear velocities. Let t\ denote the time when sliding stops, that is, when v  au. Then
+ ugh = a (co0 - ^Ll J,
nr                                               /, - ac - V
Substituting this value of t in equations (6) and (7) we get
a,                                                   (9)
and                                     L =            ,                                  '     (10)
for the linear and the angular velocities of the hoop after the instant when the sliding ceases. The subsequent motion is one of pure rolling with a linear velocity v\, greater than VQ, and angular velocity coi, less than o>0.
Case II.  Initial rotation clockwise and VQ > aco0.    In this case sliding is toward the right, consequently F is negative and therefore
 w,                                   (*">
dt      a
If fe denotes the time when sliding stops, in this case, a reasoning similar to the foregoing gives